where \(\tau _1=\tau \)  is the kissing number and \(d_1=d(\Lambda )=2\rho (\Lambda )\)  is the lattice minimum distance. It is well-known that the series \(f(\Delta )\)  converges for \(\Delta >0\) , because
the Theta series itself converges for \(|q|<1\)  and it is holomorphic in \(z\)  for \(q=e^{i \pi z}\) 
and \(\Im {z}\ge 0\)  [1]}.
Another direct method is to upperbound \(\tau _k\) , for \(k\)  large, by the number
of points on a sphere in \(\mathbb {R}^n\)  of radius \(d_k\)  where each point is occupying
an area given by a sphere in \(\mathbb {R}^{n-1}\)  of radius \(\rho \)  to prove that \(\tau _k\)  is polynomial in \(d_k\) .
The sequence \(d_k\)  is unbounded and strictly increasing, hence \(f(\Delta )\)  converges
for \(\Delta >0\) . We will be just using the fact that \(f(1)\)  is finite to prove (15).
Indeed, we can write
\(\frac{\sum _{k=2}^{\infty } \tau _k \exp (-\frac{\Delta }{8\sigma _{max}^2}\cdot d_k^2)}{\tau _1 \exp (-\frac{\Delta }{8\sigma _{max}^2}\cdot d_1^2)}&=\sum _{k=2}^{\infty } \frac{\tau _k}{\tau _1} \exp \left(-\frac{\Delta }{8\sigma _{max}^2}\cdot (d_k^2-d_1^2)\right)\\ \nonumber &=\sum _{k=2}^{\infty } \frac{\tau _k}{\tau _1} \left[\exp \left(-\frac{d_k^2-d_1^2}{8\sigma _{max}^2} \right)\right]^1\left[\exp \left(-\frac{d_k^2-d_1^2}{8\sigma _{max}^2} \right)\right]^{\Delta -1}\\ \nonumber &\le f(1) \cdot \left[\exp \left(-\frac{d_2^2-d_1^2}{8\sigma _{max}^2} \right)\right]^{\Delta -1},\) 
